Methods

Paul J Birrell; Richard G Pebody; André Charlett; Xu-Sheng Zhang; Daniela De Angelis

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Birrell PJ, Pebody RG, Charlett A, et al. Real-time modelling of a pandemic influenza outbreak. Southampton (UK): NIHR Journals Library; 2017 Oct. (Health Technology Assessment, No. 21.58.)

Real-time modelling of a pandemic influenza outbreak.

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Chapter 3Methods

We shall begin by describing the modelling approaches used in this work (see Modelling methodology: single region model and Modelling methodology: multi-region models). In Data we will examine the data types that PHE currently envisage being available at some stage during a pandemic and discuss how the structure of the available data from 2009 helped to determine the precise parameterisation of the real-time model. Together, the modelling approaches and data types will inform the spatial modelling study.

Bayesian inference provides an introduction to Bayesian inference and Monte Carlo methods discusses the MCMC and SMC methods. Monte Carlo methods, in particular, contains a significant amount of technical detail, including the tuning of a number of algorithmic components necessary to achieve timely inference, and may be omitted by the reader not interested in such detail.

Modelling methodology: single-region model

The starting point for the investigation in this study is the model and analysis of Birrell et al.¹⁵ Here, information from multiple sources is integrated into a composite model, including:

an age-structured dynamic transmission component
a disease component
a component describing the mechanism of symptom reporting to health-care facilities.

A schematic model representation is given in Figure 1. Transmission in the SEIR model is governed by a time- and age-varying force of infection that is dependent on the population structure, the transmissibility of the virus, the mixing patterns between population strata and the expected time spent in the E (exposed) and I (infectious) states. In the disease model layer, a proportion, ϕ, of the newly exposed individuals develop febrile symptoms. In the reporting model layer, further proportions of these symptomatic individuals consult their GP and/or have their symptoms officially confirmed through a virologically positive swab result.

FIGURE 1

Ideally, direct data on the number of new infections would be available, and in studies of modelling methodology these types of data are often assumed.¹¹ However, more realistically, surveillance data sets are noisy and record events (such as GP consultations; see section entitled Pandemic data) that occur some time later than the infection. Specifically for influenza, disease reporting is frequently through syndromic surveillance, whereby non-disease specific symptoms are reported. Instead of reporting influenza infections, the reporting is of patients suffering with ILI. Therefore, data from such sources include contamination from patients carrying infections other than the pathogen of interest. This adds greater complexity to the task of disentangling the underlying disease incidence from the available information, particularly as this contamination is likely to vary substantially during the pandemic. To identify the disease incidence, these noisy consultation data are combined with information on virological positivity from complimentary surveillance systems (see Figure 1 and Data). When multiple time series data sets are available, data on events occurring as close as possible to the time of infection should be preferred, as they will be more informative. Alternatively, data arising as a result of severe symptoms are also valuable; severity is a property of the virus, and so the proportion of cases that appear in data will become more stable over time.

The equations governing the epidemic dynamics are included in Appendix 1.

Modelling methodology: multi-region models

The transmission model shown in Figure 1 and Appendix 1 is extended to accommodate spatial heterogeneity in the 2009 A/H1N1 pandemic data in two ways: (1) by using a parallel-region (PR) approach and (2) by using a meta-region (MR) approach.²⁸ These approaches will be introduced in the two following subsections, respectively.

The parallel-region model

In the PR modelling approach, the spatially heterogeneous epidemic is assumed to be composed of a number of smaller epidemics occurring in parallel within each spatial unit, with no direct interaction (specifically, no transmission) between regions. The rationale here is that the purpose of the real-time model is to monitor the pandemic once infection is widespread. By such a time, it is reasonable to assume that long-range inter-region transmission will be negligible in comparison with that occurring within each region.

The parallel epidemics are still jointly modelled; however, there is a sharing of information through a number of model parameters that either have a common value or a common temporal trend in each region. These parameters are typically those representing biological characteristics of the virus (mean infectious period, proportion symptomatic, etc.). Additionally, the mixing patterns are assumed to exhibit no regional variation. Appendix 1 presents a system of equations governing single-region dynamics. This system is driven by two key quantities: the reproductive number, R₀, and the initial state of the system, defined by a parameter giving the initial number of infective individuals, I₀. These are region specific parameters (R_0,r, and I_0,r, r = 1, . . . ,R) as they are functions of both the regional population and the virus. Together, these parameters account for the different timing of the pandemic activity in each region.

The system of dynamic equations given in Equation 18, Appendix 1 applies within each spatial unit, and so needs little modification.

The meta-region model

In the MR modelling approach, regions are assumed connected such that transmission is possible between individuals resident in different regions. Here, we look at the country as a whole and treat it as a metapopulation of R sub-regions. Therefore, we can generalise the notation in the system of equations (see Equation 18) in Appendix 1 so that the index a now takes values over the range 1, . . . ,RA. It is therefore necessary to define (RA × RA) contact matrices, Π(t_k), k = 1, . . . ,K, that describe the rates at which individuals of the various (region- and age-defined) strata come into contact. In the single-region and PR models, this matrix describes the rates at which individuals of the various age groups interact, and was informed by UK data collected as part of the Improving Public Health Policy in Europe through the Modelling and Economic Evaluation of Interventions for the Control of Infectious Diseases (POLYMOD) study (see Equation 20 and relevant text).²⁹ In this expanded matrix, the entries that correspond to within-region contacts resemble the POLYMOD-based matrices. However, entries corresponding to inter-region interactions are typically of a lower order of magnitude, as people interact less frequently with others living in a different geographic region. These rates of contact are derived from census data on daily commuter movements between regions. The details of how, at the k_th time point, t_k, the POLYMOD matrices and the commuter data combine to produce contact matrices Π(t_k) are given in Appendix 1. Figure 2 shows a heat map of the elements of Π(t₁) as contact intensities on the absolute and log-scales. The strata are organised within regions, giving the matrix the appearance of an array of submatrix blocks within which the POLYMOD patterns of contact are repeated. The blocks on the diagonal give rates of within-region contact and, therefore, show much higher contact rates.

FIGURE 2

Heat maps for the contact matrices used in the MR model (with regional density dependence) based on (a) contact rates; and (b) log-contact rates. The matrices show a strong, block-diagonal structure, with red areas indicating higher rates of contact. (more...)

The MR model has only one system of dynamic equations of the type in shown in Equation 18 of Appendix 1, removing the flexibility of having region-specific values for R₀ and the initial seeding of infectives, I₀. However, there are a number of modelling considerations to be made when using the MR model, considerations that are not relevant to the PR modelling approach.

Density dependence

In a single-region model, the POLYMOD-based contact matrices give the relative frequency of contact between pairs of individuals of the different age groups. When these matrices are inserted into the block diagonal of the MR contact matrix (see Figure 2), an assumption of frequency-dependent contact is made. This implies that individuals are equally likely to have contact with any other individual in the region irrespective of the region in which they live. Therefore, individuals who live in regions with a higher population will make proportionately more contacts. This may not seem to be a reasonable assumption, as the total population of a region does not necessarily indicate a high population density. The alternative considered here is density-dependent contacts. In this case, the likelihood of a contact is scaled down by the population size, either of the regional population or of the strata (region and age) population. Both types of density dependence are considered.

Initial seeding of infectives

At the beginning of an epidemic, the transmission process is kick-started by a number of initial infectives. The POLYMOD-based contact matrices used for a single-region model (and in the PR model) will lead to rapid convergence towards a stable pattern of infection. In other words, for most reasonable choices of initial seeding, it takes only a short time for this seeding to be ‘forgotten’ by the transmission dynamics.

This is not the case in the MR model, with its block-structured matrix. Infection spreads very slowly between the regions, and so the initial seeding is not quickly forgotten. Epidemics seeded with infectives in different regions can lead to very different outcomes. Therefore, although the choice of the seeding in the PR model is not important, it is a significant modelling choice for the MR approach and a number of different seedings are considered: the single-region equilibrium distribution (i.e. the stable pattern of infection observed in the PR model) derived from the next generation matrix; the initial empirical age- and region-specific distribution of the initial confirmed cases; and a hybrid approach using a within-region equilibrium distribution scaled by the empirical distribution of the initial confirmed cases over regions.

Commuting at random

Each model makes the assumption of homogeneous mixing within each stratum. This means that all individuals within a population stratum are as likely to acquire or spread infection as any other individual of the same infection status. In the MR model, where infection is transmitted between regions through the routine movements of commuters, this assumption of homogeneity implies that on any day each individual is equally likely to commute. This is an unrealistic representation, however, as only a subset of people are likely to commute on a regular basis, and a (larger) subset are likely to stay within their home region. To account for this, adult age groups in each region are further subdivided into commuters and non-commuters, so that the commuters are a fixed group of people who move each day. This results in the effects of commuting on transmission being more transient, as there is a smaller, more rapidly exhaustible supply of susceptible individuals available to transfer infection from one region to another.

The downside to this further level of stratification is that it places an increased computational burden on the model, greatly slowing down the estimation process.

Data

During the 2009 pandemic, the HPA provided A/H1N1pdm incidence estimates for each of the then ten Strategic Health Authorities (SHAs) across England. Two of the SHAs, Greater London and the West Midlands, were believed to have experienced a significant, pre-summer wave of infection. Ideally, we would adopt the same geographical partition; however, the volume of data available from each of the SHAs is insufficient to do so. A reasonable compromise solution is to divide the country into four spatial units: London, the West Midlands (the two regions that had significant first waves of infection), the North and the South. The North and South regions each comprise four SHAs; in the North these are North East, North West, Yorkshire and Humberside, and East Midlands, and in the South they are East of England, South Central, South East Coast and South West.

The age categorisation favoured by the HPA, now PHE, is to break up the population into the age groups < 1 year, 1–4 years, 5–14 years, 15–24 years, 25–44 years, 45–64 years and ≥ 65 years. For the rest of this section, a will denote a general population stratum, whether defined by age alone or by both region and age.

The section, Pandemic data below, itemises the pandemic data streams that the real-time modelling framework is set up to work with, detailing, when applicable, how these data were used in modelling the 2009 pandemic and discussing how the various surveillance schemes have evolved over the intervening period. Distributional assumptions introduces some technical statistical details, showing how these data streams link into the SEIR transmission model, covering the distributional assumptions that are required to allow formal statistical inference to be made.

Pandemic data

General practice consultation data

Public Health England carry out syndromic surveillance to monitor influenza activity in the population by routinely collecting data on individuals presenting with an ILI at GPs. In 2009, such data were provided from two sources. The first source was the Weekly Returns Service of the RCGP, a sentinel GP network covering a weekly population of approximately 900,000.³⁰ The second source was the HPA/QSurveillance national surveillance system, which covers a much larger population of ≈23 million people.³¹ ILI data from both schemes were available stratified by both age group and SHA. In the end, daily ILI reports from the QSurveillance system were used to guide the public health response to the pandemic.³¹

The GP data are reported counts of consultations for non-pandemic specific symptoms, and include cases not infected with the pandemic pathogen. Therefore, information is required on the proportion of the reported counts that are truly of interest when tracking the levels of transmission of the pandemic infection. The RCGP augment their primary care surveillance with virological monitoring.³² This monitoring involves taking respiratory swabs from a subset of patients (chosen at random) consulting for ILI at participating GPs. A polymerase chain reaction (PCR) assay is then employed to test the swabs for the presence of influenza strains as well as for other respiratory virus infections. Similar data are obtained and made available by PHE’s RMN, covering an additional 400,000 patients in England.³³ A complete account of the virological monitoring undertaken by the two schemes throughout the 2009 pandemic can be found elsewhere,³⁴ but together they provide data on the positivity of the swabs taken in GPs together with the epidemiological information attached to each sample. To ensure high sensitivity of the testing process, swabs were only included in any analysis presented here if the time between symptom onset and the swab being taken was ≤ 5 days. Combining this swabbing information with the GP consultation data, the number of consultations that are actually directly due to the pandemic can be estimated.

Since 2009, PHE has expanded its primary care surveillance portfolio, now additionally working with The Phoenix Partnership to access anonymous GP records through its SystmOne™ computer system (The Phoenix Partnership, Leeds, UK).³⁵ These data could either be combined with the QSurveillance data or provide an additional sample of data used for model validation. When infection becomes widespread, the National Pandemic Flu Service (NPFS) will be activated. The NPFS is an internet and telephone service designed to expedite the administration of antiviral drugs, alleviating the burden placed on GP surgeries. This service was launched in 2009, and, after a short bedding-in period, was observed to be subject to the same trends as the GP-based data. Those using the service were also swabbed, to understand the underlying pandemic incidence. Owing to an anticipated fall in the consultation numbers that would arise as a result of a NPFS launch, these data could easily be added (if the degree of overlap between the two data sets is understood) or used to replace the GP consultation data in order to build a picture of the numbers accessing primary care services for ILI.

Virologically continued cases

Management strategies over the initial stages of the 2009 pandemic were primarily concerned with containing the spread of the epidemic, before moving into a treatment phase. The initial containment phase was a period of enhanced surveillance during which contacts of known infected individuals were traced and laboratory confirmations of the infection were obtained whenever possible. This work resulted in the generation of the first few hundred (FF100) and the FluZone databases.³⁴ Routine laboratory confirmations were discontinued on 25 June 2009, but we use here the data only up to 19 June 2009 to allow for the gradual cessation in the collection of this type of data. In practice, any real-time modelling is likely not to start within the first 5 weeks of the outbreak, given the anticipated difficulty in detecting any signal from the epidemic data at such early stages. Instead, it is anticipated that the information on confirmed cases in this period will inform the model construction and provide some prior information (see Model parameterisation) for various model parameters. In the analysis of the 2009 pandemic data, these data contributed to the analysis in the same way it is proposed that hospitalisation data will in future pandemics (see Hospitalisation data – UK Severe Influenza Surveillance System and Distributional assumptions).

Hospitalisation data – UK Severe Influenza Surveillance System

Prior to the 2009 pandemic, there was a gap in the surveillance of severe respiratory infections in the UK with regard to hospitalised cases of influenza. During the pandemic, a web-based hospital reporting system was established to meet this need. The data were available relatively late in the pandemic and, even now, the biases and weaknesses of the data derived from this reporting system are not well understood. This motivated the development of a more robust, well-tested surveillance scheme for the reporting and handling of such important data. As a result, the UK Severe Influenza Surveillance System (USISS) was initiated during the 2010/11 influenza season, becoming routine for each subsequent season.³⁶ Data collected during influenza seasons prior to any pandemic outbreak are anticipated to provide baseline information that may prove useful in identifying a pandemic ‘signal’.

Outside a pandemic, USISS is a two-stream surveillance system. All NHS hospital trusts carry out mandatory weekly reporting of admissions of severe influenza cases (i.e. patients admitted to a high-dependency unit or an intensive care unit (ICU) with laboratory confirmation of infection). USISS also provides sentinel influenza surveillance through an annually selected random sample of trusts, where testing for the presence of influenza in all patients presenting with ILI is mandatory, and results are reported together with an array of epidemiological information. In the event of a pandemic being declared, all trusts will switch to this level of reporting.

In a pandemic, therefore, USISS should provide a time series of reported cases that have two distinct advantages over the GP consultation data:

They are counts of laboratory confirmed cases, so there is no contamination from non-pandemic ILI.
The proportion of cases that are reported to USISS is a function of the severity of the virus and access to hospital services, and should therefore be less volatile over time.

However, hospital resources are finite, and, in a rapidly developing pandemic, may quickly become exhausted, so that patients may be turned away where previously they would have been hospitalised. In such a situation, the proportion of patients who are hospitalised will decrease as a function of the increasing incidence. This decrease may be difficult to characterise, potentially limiting the period of time for which the hospitalisation data can be reliably informative.

Serological data

Serological data are the only surveillance data source directly informing the transmission component of the real-time model. As the prevalence of immunity-conferring antibodies increases, the number of susceptibles decreases. In modelling the 2009 pandemic, the inclusion of serological data has been shown to be crucial to the reconstruction of the underlying epidemic curve.¹⁵

Initially, the serological data used in the analysis of the 2009 data came from the HPA’s annual collection of residual blood serum samples submitted to microbiological laboratories for the purpose of carrying out cross-sectional antibody prevalence studies.³⁷ Later in the pandemic, it became clear that a more rapid, more representative approach to the collection of serum samples was required. Chemical pathology laboratories were therefore approached at hospitals in each of the RMN regions. This ensured a regular supply of age-stratified serum samples, obtained in a timely fashion and with good geographical coverage.³⁸ In all samples, a haemagglutin-inhibiting antibody titre of 32 was assumed to be sufficient to indicate protection against A/H1N1pdm influenza.³⁹^–⁴¹ It is further assumed that there is a 2-week delay between infection and seroconversion. Each sample was, therefore, treated as representative of the level of cumulative infection among the population 14 days prior to the sampling date. Testing of some residual sera samples collected in 2008 also took place to provide age-specific estimates of baseline antibody prevalence.

In recent years, ahead of each winter influenza season, researchers at PHE have carried out stratified sampling from the population to select potential participants for a telephone survey regarding the public’s attitudes towards influenza vaccination.³⁶ At the end of the survey, respondents were asked if they would be willing to submit a blood serum sample. Those who agreed to take part submitted two samples, one at the start of the season and one at the end of the season. In the event of a pandemic outbreak that does not overlap with the winter flu season, the telephone surveys will be reactivated as rapidly as possible.

There is some uncertainty inherent in these data as to precisely what titre value will confer immunity. It is also possible that, to indicate long-standing immunity, different titre levels may be required from levels that indicate recent infection. The real-time modelling system does allow for this potential difference in the titre thresholds, but it does not yet account for any uncertainty in these values.

Commuting data

Commuting data were extracted from the UK 2001 census.⁴² For individuals aged ≥ 16 years, these data are in the form of counts of the number of surveyed individuals in each age group and within each Government Office Region (GOR) who, on the day of the census, travelled into another GOR and the number who stayed within their home region. Data were then aggregated so that they conformed to the regional split chosen for modelling the 2009 pandemic: London, West Midlands, the North and the South. The number of people in age group a who moved on the day of the census from region r to region s are denoted by $C_{r, s}^{*} (a)$ , and these numbers were standardised to give:

C_{r, s} (a) = \frac{C_{r, s}^{*} (a)}{\sum_{u = 1}^{R} C_{r, u}^{*} (a)} .

(1)

Equation 27 in Appendix 1 illustrates how these data are combined with information from the UK component of the POLYMOD study to generate contact matrices suitable for use in the MR approach to handling spatially heterogeneous epidemics.

Population totals stratified by agegroup and GOR were also derived from UK Office for National Statistics (ONS) data, using the 2008 mid-year estimates.⁴³

Distributional assumptions

Count data

General practice consultation data, virological confirmations and hospital admissions are all examples of count data that the real-time modelling framework has been designed to accommodate. We assume that these are realisations of either Poisson or Negative Binomial distributions. The expectations of these distributions have derivations that share some common features, accounting for:

A delay from infection to the health-care event being recorded.
The fact that these are a proportion of the symptomatic cases: those with a sufficiently severe illness or who make a particular health-care choice. For data on hospitalisations, this proportion is the case-hospitalisation or case-ICU risk, and, for GP consultations, it is a time-evolving propensity for individuals to seek consultation in the presence of symptoms.

Therefore, the expected number of daily reports of hospitalisations, denoted $µ_{a}^{h} (t_{k})$ for day t_k within stratum a is linked to the daily number of new infections through an expression of the type:

E [X_{k, a}^{h}] = µ_{a}^{h} (t_{k}) = p_{a}^{h} (t_{k}) \sum_{l = 0}^{k - 1} q_{l}^{h} Δ_{a}^{(infec)} (t_{k - l}),

(2)

where $p_{a}^{h} (t_{k})$ is the relevant case-severity risk, $q_{l}^{h}$ is the probability that the time taken from infection to being reported in data as having been hospitalised spans l time intervals, and $Δ_{a}^{(infec)} (t_{k})$ is the number of new infections at time t_k as found from Equation 21 in Appendix 1.

For GP surveillance data, the expected number of consultations arising from the pandemic, $µ_{a}^{g} (t_{k})$ is calculated via a similar expression to Equation 2:

µ_{a}^{g} (t_{k}) = p_{a}^{g} (t_{k}) \sum_{l = 0}^{k - 1} q_{l}^{g} Δ_{a}^{(infec)} (t_{k - l}) .

(3)

Before this quantity can be related to data, however, there are three additional considerations to make:

The non-pandemic consultations need to be added. These consultations are denoted by B_a(t_k), the expected values of these at time t_k.
The within-week pattern of consultations has to be accounted for. Typically, no data are reported at weekends and on bank holidays. This leads to a strong artefactual peak in the number of consultations each week on Mondays.
Although the population coverage of PHE’s combined surveillance schemes in England is very high, it is still incomplete, and this needs to be accounted for. The expected number of consultations needs to be scaled to allow for this incomplete coverage. Surveillance schemes will report daily coverage figures as a proportion of the total population in each stratum, which we denote as D_a(t_k).

The expected daily counts of consultations in a general stratum a on day t_k are $E [X_{k, a}^{g}]$ such that:

E [X_{k, a}^{g}] = D_{a} (t_{k}) κ_{d (t_{k})} (µ_{a}^{g} (t_{k}) + B_{a} (t_{k})),

(4)

where d(t_k) indicates the day of the week on which time t_k falls and κ_{d(t_k)} is the adjustment factor accounting for the within-week effects on reporting. These factors should be estimated subject to the constraint that $\prod_{d = 1}^{7} κ_{d} = 1.$

In particular, the GP data are likely to be highly volatile owing to the sensitivity of the population’s health-care-seeking behaviour to governmental advice and media reporting. If it is decided that, as in 2009, the most appropriate distribution for the consultation data is the negative binomial, then the real-time model will include dispersion parameters, $η_{k, a}^{g}$ , such that the variance is given by:

Var [X_{k, a}^{g}] = (1 + η_{k, a}^{g}) E [X_{k, a}^{g}] .

(5)

Sampling data

Both the virological and serological data represent a number of positive readings in a sample of fixed size.

We denote the virological data as $(m_{k, a}^{v}, W_{k, a})$ where $m_{k, a}^{v}$ gives the number of swabs tested within 5 days of symptom onset and W_k,a is the number of those swabs that test positive for the presence of the pandemic pathogen. If we assume that the PCR test has test sensitivity k_sens and test specificity k_spec, then these data are binomially distributed with the expected value:

E [W_{k, a}] = m_{k, a}^{v} (k_{sens} \frac{µ_{a}^{g} (t_{k})}{µ_{a}^{g} (t_{k}) + B_{a} (t_{k})} + (1 - k_{spec}) \frac{B_{a} (t_{k})}{µ_{a}^{g} (t_{k}) + B_{a} (t_{k})}) .

(6)

In all of the analyses presented in this report, the virological testing procedure is assumed to be perfect, with

k_{sens} = k_{spec} = 1 .

(7)

Similarly, we denote the number of blood sera samples that test positive for the presence of antibodies as Z_k,a among a total of $m_{k, a}^{s}$ samples. The expected number of positive samples is linked to the level of susceptibility in the population, S_a(t_k), via:

E [Z_{k, a}] = m_{k, a}^{s} (1 - \frac{S_{a} (t_{k - k_{0}})}{N_{a}}),

(8)

where k₀ is a time lag representing the number of time-steps required for the development of antibodies. In 2009, this was taken to correspond to 14 days. If pre-season sampling occurs prior to the chosen t₁, for modelling purposes these samples can be assumed to be informative about the population prevalence of antibodies on the first day of the outbreak, and can be added as data at this time.

Model parameterisation

Apart from parameters that describe some initial condition of the transmission model, parameters are permitted to vary over time, region and age. Appendix 2 details all the model parameters that can, in principle, be estimated within the real-time model framework. In reality, depending on the availability of relevant data, a subset of parameters is pragmatically chosen for estimation. Table 1 presents a list of the parameters estimated in the spatial analysis of the 2009 pandemic data, indicating whether each parameter varies across regions (denoted as ‘spatial’) or not (denoted as ‘global’).

TABLE 1

Model parameters classified in the PR and MR models as either spatially varying or globally varying

Parameters d_l and ɸ are deemed to be properties of the virus and therefore are treated as constant over region, time and age. Parameters ψ_r and v_r describe initial conditions (see Appendix 2 for their interpretation) and therefore they have a region-specific value in the PR model and a global value in the MR model. As virological case confirmation data were used in the absence of consistent data on hospitalisations, the proportion of cases that received virological confirmation of their infection, here set to be p^h, is an observation model parameter, relevant only for the first 50 days of the epidemic while this type of data were still being collected. Therefore, no temporal or age-specific variation is considered, although variation over regions is included on account of the very different levels of pandemic activity in each region over the early period. The specification of the parameter κ_d has already been discussed in the text following Equation 4.

Parameter vector m = (m₁ , . . . , m₅) consists of mulitpliers to specified elements of the contact matrices. These parameters are used to measure the impact of school holidays on contacts among 1- to 4-year-olds and 5- to 14-year-olds, and to down-weight the contribution of all contacts involving at least one adult.

Both parameter vectors η and p^g are properties of the reporting model for the GP consultations. Therefore, they both have a temporal change point at time t_k = 83 days, the time of the NPFS launch. Additionally, p^g differs across ages (different values for children and adults), as well as changing value at two points later in the epidemic to account for the gradual reversion in the public’s health-care-seeking behaviour to pre-NPFS habits. Thus, η is an eight-dimensional parameter component and the p^g parameters are 32-dimensional.

The parameters describing the rates of non-pandemic ILI consultations, known as the background rates of consultation, have the most complex specifications. Regional variation in these rates is specified through a log-linear regression model, allowing information on trends and age effects to be shared across regions. As the background rates of consultation are quite likely to be volatile over time, approximately fortnightly breakpoints are chosen, dividing the 245 days under study into 17 distinct time segments.

The modelling process begins with a first-phase of model choice within the PR modelling framework to specify the precise form of this regression. Letting τ(t_k) denote the fortnightly interval into which time t_k falls, and explicitly denoting the strata (r,a), a saturated model for the consultation rates, B_r,a(t_k), takes the form:

log (B_{r, a} (t_{k})) = µ + α_{r} + β_{τ} + γ_{a} + δ_{r τ} + ε_{r a} + ζ_{τ a} + τ_{r τ a}; r ε {L, W, N, S}, τ = 1, . . ., T_{X}, a = 1, . . ., A .

(9)

log (B_{r, a} (t_{k})) = µ^{*} + α_{r}^{*} + β_{τ}^{*} + γ_{a}^{*} + δ_{r τ}^{*} + ε_{r a}^{*} + ζ_{τ a}^{*} + τ_{r τ a}^{*}; r ε {L, W, N, S}; τ = T_{X} + 1, . . ., T; a = 1, . . ., A .

(10)

Parameters in Equations 9 and 10 represent the main effects of region (r), time period (τ) and age group (a) and their interactions; T_X indicates the fortnightly time interval that concludes at the same time as the launch of the NPFS, and the regions {L, W, N, S} correspond to London, West Midlands, the North and the South, respectively. This specification implies that there are separate and non-interacting models for the periods pre- and post-NPFS launch. In a preliminary phase of modelling, regression terms from Equations 9 and 10 are sequentially removed until there is an appreciable loss of fit to the data, to obtain simplified versions of the regression equations (see Chapter 4, section Finding an optimal parameterisation).

Bayesian inference

In the Bayesian framework, statistical inference about an unknown parameter of interest, θ, proceeds by combining a priori information about θ with data from a current study. The initial information on θ is expressed in terms of a probability distribution, p(θ), known as a prior. This distribution encapsulates all that is known (or not known) about the parameter (e.g. from expert opinion or historical data) before the current study is carried out. After carrying out the study and observing data y, the knowledge about parameter θ is updated to give a probability distribution, known as the posterior, p(θ|y). This posterior distribution is found through Bayes’ formula:

p (θ | y) \propto L (y; θ) p (θ),

(11)

where L(y; θ) is the likelihood function, expressing the likelihood of observing data y conditional on the parameter taking value θ. The likelihood for the spatial study and a summary of the chosen prior distributions for the parameters are given in Likelihood and Priors, below.

Likelihood

Denoting the number of days over which we have epidemic data as K, and using bold to denote (possible) vector quantities, we write that the epidemic data set is y_1:K = (y₁,. . .,y_K). Each data vector y_k contains components $(w_{k}, x_{k}^{g}, x_{k}^{h}, z_{k})$ , consisting of the data types discussed in Distributional assumptions, with each data component containing stratum-specific data reported at time t_k.

These data contribute to inference through the likelihood function. Conditional on all the model parameters, using θ to denote the parameters in Table 2, it is assumed that all observations can be considered independent. The likelihood is then expressed as:

TABLE 2

Prior information on model parameters. For each parameter grouping, the table specifies the prior distribution used, or, where the parameter is not to be estimated by the model, its fixed value

L (y_{1 : K}; θ) = \prod_{k = 1}^{K} \prod_{a = 1}^{R A} {L (w_{k, a} | m_{k, a}^{v}, θ) L (x_{k, a}^{g} | θ) L (x_{k, a}^{h} | θ) L (z_{k, a} | m_{k, a}^{s}, θ)} .

(12)

The terms inside the product correspond to the likelihood of virological swabbing data, GP consultation data, USISS hospitalisation data and serological data, all reported at time t_k and for each stratum a (assuming here that this encompasses both age group and region).

Priors

Table 2 provides a summary list of model parameters and, in the spatial analysis, their assumed prior distributions, or fixed (known) values as applicable. In some rows of the table, dependence on region has been made explicit through the use of a subscript r.

The justifications for the majority of the choices in the table have been given elsewhere¹⁵ and Model parameterisation outlines which parameter components have been considered for the additional spatial variation. When regional variation exists, parameters are identically distributed in each region.

In short, parameters that are hard to estimate from this type of model and surveillance data, such as d_I and ϕ, have informative prior distributions based on historical studies and analyses of early epidemic data.¹⁵ The prior for p^g uses information from FluSurvey,⁴⁴ whereas the priors on the parameters ψ_r and ν_r can be considered to be relatively uninformative, and the priors for the components of η are particularly diffuse. The bottom two rows of Table 2 are for parameters used to calculate the background consultation rates and the day-of-the-week effects. These are given zero-mean multivariate normal distributions with covariance matrices V^B and V^K. The covariance matrices are designed so that the background quantities B_r,a(t_k) and κ_d, d = 1, . . . ,7, are uncorrelated and identically distributed wherever possible.

Monte Carlo methods

Typically, the posterior distribution of Equation 11 is known only up to a constant of proportionality and, as a result, is seldom possible to derive analytically, particularly when working with a model as complex as that of Figure 1. However, it is possible to obtain a sample from such a distribution. The class of methods used to produce such a sample are called Monte Carlo methods, and two of the most common such methods are discussed below.

Markov chain Monte Carlo

Markov chain Monte Carlo (MCMC), a widespread and popular algorithm for Bayesian computation, is used to derive estimates of the posterior distribution of the model parameters in the spatial analysis of 2009 pandemic data. More detailed introductions to MCMC can be found elsewhere;²¹ however, in short, MCMC techniques are used when it is necessary to sample from a distribution when this sampling cannot be done directly. In any complex modelling scenario, the posterior distribution in Equation 11 represents such a distribution. MCMC works by generating a sequence of values known as a Markov chain. If allowed to run for a long time, this chain will eventually constitute a dependent sample from the desired distribution. Typically, one would run a small number of such chains (approximately between two and five), starting each chain at dispersed values. The chains run for a burn-in period until samples derived from each are statistically similar. At this point, it can be said that the chains have converged, and then the chains run for a sufficient length of time to derive a sample of the desired quality. This can often require many iterations of the chain (in applications of dimension comparable with the dimension of the parameter vector in our example, often 10⁴−10⁶ iterations may be required) and can often be a time-consuming process as a result.

To see how to generate a sample from a posterior distribution, we introduced some technical detail outlining the Metropolis–Hastings (MH) algorithm, the most commonly used MCMC method. Formally suppose that, at time t_k we are trying to derive a sample from the posterior p(θ|y_1:k), where y_1:k denotes all of the data observed up to the present time. Suppose the parameter value at the n^th iteration of the chain is θⁿ. From a carefully chosen probability distribution known as the proposal distribution, a new state for the chain is proposed, θ*∼q_k(·|θⁿ). This value is then accepted as the next state of the chain with the probability:

min (1, \frac{p (θ^{*} | y_{1 : k}) q_{k} (θ^{n} | θ^{*})}{p (θ^{n} | y_{1 : k}) q_{k} (θ^{*} | θ^{n})}) .

(13)

If the proposed parameter value is not accepted, then the chain stays where it is and θⁿ⁺¹ = θⁿ.

The performance of a MCMC algorithm crucially rests on the choice of the proposal distributionzs q_k(·|·). However, regardless of this choice, the algorithm remains highly linear, with minimal scope for taking advantage of the benefits offered by parallel computing. Therefore, this algorithm will struggle to reap any of the benefits of cluster computing. More importantly, in an iteration of the algorithm, the suitability of proposed values is evaluated using knowledge of the full data likelihood (from time t₀ to time t_k). This will require the evaluation of the system of equations in Equation 18 in Appendix 1 and of Equations 2 and 3. When repeated 10⁵ (orders of magnitude) or more times, this can compromise the capacity for timely, real-time inference.

This motivates a more readily parallelisable algorithm, and one that is sequential in nature, demanding only the evaluation of the likelihood of the incoming batch of data, rather than the full data history.

Sequential Monte Carlo

In general terms, SMC provides a prescription to sample from a target probability distribution, denoted π(.), by sequentially moving through a number of, say, L intermediate distributions π₀(·), . . . , π_L(·) = π(·). By setting K = L and π_L(·) = p(·|y_1:L), it can be seen how this algorithm may lend itself to the problem of online inference. At the k^th stage of the sequence of target distributions, a weighted sample of size n_k from p(·|y_1:k) is obtained, denoted:

{(ω_{k}^{(1)}, θ_{k}^{(1)}), . . ., (ω_{k}^{(n_{k})}, θ_{k}^{(n_{k})})} .

(14)

Here, the weight $ω_{k}^{(j)}$ attached to a parameter value $θ_{k}^{(j)}$ , known in this context as a particle, indicates the relative importance of the j^th particle to the sample (known as the particle set). This means that if we have a function of the parameter, such as the epidemic trajectory, which we denote as ƒ(θ), then we would estimate it by its weighted mean:

\frac{1}{n_{k}} \sum_{l = 1}^{n_{k}} ω_{k}^{(j)} ƒ (θ_{k}^{(j)}) .

(15)

The basic idea is that the SMC algorithm proceeds, on observing a (k + 1)^th batch of data, by reweighting the sample according to the likelihood of the new data. This reweighted sample is theoretically representative of the next target distribution π_k + 1(·)=p(·|y_1:(_k + 1)). Therefore, we can base inferences at time t_k + 1 on the previous sample of parameter values and the new set of weights, which require only the likelihood of the new data to be calculated. This represents a significantly reduced computational burden, and, as the reweighting for each of the particles can be calculated in parallel, it is also a highly parallelisable computation.

Unfortunately, such a process swiftly suffers from a phenomenon called particle degeneracy. This happens gradually over time as the particle weights scale in such a way that only a very small handful of particles have non-negligible weight. When this degeneracy occurs, although the weighted sample is of size n_k, the low weight attached to the majority effectively removes them from the sample, and estimation and projections are made based on only a handful of particles and are, therefore, subject to significant error.

To prevent this degeneracy, the sample requires some rejuvenation.⁴⁵ The first step of this rejuvenation involves the removal of all those particles of too low weight. This is done through a process of resampling, in which a new sample is drawn from the old set of particles according to their weights. The consequence of doing this is that the sample is composed of multiple copies of a much smaller number of identical particles. It is therefore necessary to jitter this sample somehow. To do this, short MCMC implementations are run for each particle, using the current value of the particle as the starting state for the chain.

The sequential Monte Carlo algorithm

A brief overview of the SMC algorithm is shown below, based on the resample-move algorithm.⁴⁶

Set k = 0. At time t₀, draw a sample ${θ_{0}^{(1)}, . . ., θ_{0}^{(n_{0})}}$ from the prior distribution, π₀(θ), set the weights $ω_{0}^{(j)} = 1 / n_{0}$ for all $j = {1, . . . n_{0}}$ .
Set k = k+1. Observe a new batch of data y_k. The particles are reweighted according to the likelihood of the incoming data.
${\tilde{ω}}_{k}^{(j)} \propto ω_{k - 1}^{(j)} L (y_{k}; θ_{k - 1}^{(j)}) .$
Has the particle set become degenerate? If not, set $θ_{k}^{(j)} = θ_{k - 1}^{(j)}, ω_{k}^{(j)} = {\tilde{ω}}_{k}^{(j)}, n_{k} = n_{k - 1}$ and return to step 2.
Resample. Choose n_k and sample
${{\tilde{θ}}_{k}^{(1)}, . . ., {\tilde{θ}}_{k}^{(n_{k})}}$
from the set of particles ${θ_{k - 1}^{(1)}, . . ., θ_{k - 1}^{(n_{k - 1})}}$ with probabilities proportional to ${{\tilde{ω}}_{k}^{(1)}, . . ., {\tilde{ω}}_{k}^{(n_{k - 1})}}$ . Re-set $ω_{k}^{(j)} = 1 / n_{k} .$
Move. For all j, move ${\tilde{θ}}_{k}^{(j)} t o θ_{k}^{(j)}$ via a short MCMC chain. If k < K return to step 2, otherwise end.

Despite the presence of the short MCMC runs in step 5, this still presents a significant improvement over the plain original MCMC algorithm because:

The computationally intensive steps of the algorithm (steps 2 and 5) both allow for calculations on each particle to be made in parallel.
At most times the particle set will not be degenerate, and hence only the likelihood of the new data needs to be calculated to reweight the sample.
Each of the many parallel MCMC chains can be assumed to start from a point that is sampled from the target distribution. There is, therefore, no need to allow the chain time to reach convergence, and only very low numbers of iterations will be required.
Before the MCMC phase starts, we already have an estimated sample from the target distribution by using the weighted sample achieved in step 2. This estimate can be used to construct good proposal distributions for the MCMC, improving its efficiency.

A number of algorithmic tweaks have been required to make the algorithm robust to the vagaries of epidemic data and characteristics of the real-time model. The technical detail involved has been presented elsewhere,⁴⁷ and only a brief overview is given here.

For how long should the Markov chain Monte Carlo run?

The MCMC chains should be run for long enough to have a rich sample of parameter values, but for no longer than is strictly necessary to maintain real-time efficiency. At the start of the MCMC phase, there may be many particles with the same parameter value, which can define a cluster. The MCMC should be run for long enough that particles from different clusters have fully intermingled and the original clusters are no longer identifiable.

To formally measure the dispersal of the particles, an estimate of the intraclass correlation coefficient (ICC) is used.⁴⁸ This measures the clustering in the value of a summary quantity calculated for each particle. The chosen summary was the projected epidemic ‘attack rate’ (the total cumulative incidence measured as a proportion of the total population).⁴⁷ At the start of the MCMC phase, the ICC = 1. As the iterations progress, this value will gradually fall and the MCMC iterations will stop once the ICC falls below a pre-defined limit. It has been shown elsewhere that values in the range of 0.1–0.2 should be adequate.⁴⁷

Choosing good Markov chain Monte Carlo proposals

Equation 13 gives the acceptance probability for a proposed value for the next state of the chain. Within SMC, the aim is to rapidly diversify the set of particles, without running the chain for too long. To do this, we want to propose values for θ* that are not too close to the current values, and that are likely to be accepted. By setting $q_{k} (θ | θ_{k}^{(n)}) = p_{k} (θ | y_{1 : k})$ , the acceptance ratio would always be 1 (so the proposal will definitely be accepted). This has the added advantage that the proposal is independent of the current state of the chain, so the set of particles would be immediately intermingled.

Unfortunately, we cannot sample directly from p_k(θ|y_1:k) so easily. However, after step 2 of the algorithm, we have a weighted sample that should approximate a sample from this distribution. Therefore, by choosing q_k(·|·) to be a multivariate normal distribution centred on the weighted mean and weighted covariance of the particle set calculated at the end of step 2, we have a distribution that approximates the target density and should ensure reasonable rates of acceptance while rapidly replenishing the particle set. This works well, provided that the particle set at the end of step 2 has not become impoverished to a degree that it cannot provide a reasonable approximation to the target distribution.⁴⁷

When to rejuvenate? When are the particles degenerate?

The standard approach is to rejuvenate the particle set when the effective sample size (ESS) falls below a specific level.⁴⁹ The ESS is a measure of the number of independent, equally weighted, observations from the target distribution that are as informative as the weighted particle set. At the end of step 2 of the algorithm, the ESS is calculated as:

ESS ({{\tilde{ω}}_{k}^{(1)}, . . ., {\tilde{ω}}_{k}^{(n_{k - 1})}}) = \frac{{(\sum_{l = 1}^{n_{k - 1}} {\tilde{ω}}_{k}^{(l)})}^{2}}{\sum_{j = 1}^{n_{k - 1}} {({\tilde{ω}}_{k}^{(j)})}^{2}} .

(16)

Values of the ESS that are close to n_k – 1 indicate a sample that contains plenty of information about the posterior distribution. A typical level above which the ESS is deemed to be acceptable (and, thus, there is no need to rejuvenate the sample) is ESS ≥ n_k – 1/2.

In some of the examples, such as those considered in section Comparison of the real-time performance of the Monte Carlo methods (see Chapter 4), there are times when the addition of the next batch of data in the sequence can lead to a sudden drop in the ESS to very low values. In such cases the MCMC algorithms have too much work to do to adjust the sample and timely inference would not be possible. Therefore, to limit the depletion in the ESS, we introduce rejuvenation steps at intermediate times, between t_k and t_k + 1, by adding in the data fractionally. If we add a fraction of the data, α, such that 0 < α ≤ 1, then a value α₀ can be identified such that the ESS falls to only approximately n_k/2. The next batch of data to arrive is either the remaining portion of the time t_k + 1 data, or a further portion of it, sufficient to once again bring the ESS down to the threshold value. This is the ‘real-time’ algorithm presented in the technical publication reporting this work.⁴⁷

Copyright © Queen’s Printer and Controller of HMSO 2017. This work was produced by Birrell et al. under the terms of a commissioning contract issued by the Secretary of State for Health. This issue may be freely reproduced for the purposes of private research and study and extracts (or indeed, the full report) may be included in professional journals provided that suitable acknowledgement is made and the reproduction is not associated with any form of advertising. Applications for commercial reproduction should be addressed to: NIHR Journals Library, National Institute for Health Research, Evaluation, Trials and Studies Coordinating Centre, Alpha House, University of Southampton Science Park, Southampton SO16 7NS, UK.

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